Theorem mexval 35960

Description: The set of expressions, which are pairs whose first element is a typecode, and whose second element is a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mexval.k	⊢ 𝐾 = (mTC‘𝑇)
mexval.e	⊢ 𝐸 = (mEx‘𝑇)
mexval.r	⊢ 𝑅 = (mREx‘𝑇)

Assertion

Ref	Expression
mexval	⊢ 𝐸 = (𝐾 × 𝑅)

Proof of Theorem mexval

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mexval.e	. 2 ⊢ 𝐸 = (mEx‘𝑇)
2		fveq2 6881	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mTC‘𝑡) = (mTC‘𝑇))
3		mexval.k	. . . . . 6 ⊢ 𝐾 = (mTC‘𝑇)
4	2, 3	eqtr4di 2814	. . . . 5 ⊢ (𝑡 = 𝑇 → (mTC‘𝑡) = 𝐾)
5		fveq2 6881	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = (mREx‘𝑇))
6		mexval.r	. . . . . 6 ⊢ 𝑅 = (mREx‘𝑇)
7	5, 6	eqtr4di 2814	. . . . 5 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = 𝑅)
8	4, 7	xpeq12d 5692	. . . 4 ⊢ (𝑡 = 𝑇 → ((mTC‘𝑡) × (mREx‘𝑡)) = (𝐾 × 𝑅))
9		df-mex 35945	. . . 4 ⊢ mEx = (𝑡 ∈ V ↦ ((mTC‘𝑡) × (mREx‘𝑡)))
10		fvex 6894	. . . . 5 ⊢ (mTC‘𝑡) ∈ V
11		fvex 6894	. . . . 5 ⊢ (mREx‘𝑡) ∈ V
12	10, 11	xpex 7751	. . . 4 ⊢ ((mTC‘𝑡) × (mREx‘𝑡)) ∈ V
13	8, 9, 12	fvmpt3i 6995	. . 3 ⊢ (𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅))
14		xp0 5761	. . . . 5 ⊢ (𝐾 × ∅) = ∅
15	14	eqcomi 2770	. . . 4 ⊢ ∅ = (𝐾 × ∅)
16		fvprc 6873	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = ∅)
17		fvprc 6873	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mREx‘𝑇) = ∅)
18	6, 17	eqtrid 2808	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅)
19	18	xpeq2d 5691	. . . 4 ⊢ (¬ 𝑇 ∈ V → (𝐾 × 𝑅) = (𝐾 × ∅))
20	15, 16, 19	3eqtr4a 2822	. . 3 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅))
21	13, 20	pm2.61i 184	. 2 ⊢ (mEx‘𝑇) = (𝐾 × 𝑅)
22	1, 21	eqtri 2784	1 ⊢ 𝐸 = (𝐾 × 𝑅)

Syntax hints:  ¬ wn 3   = wceq 1568   ∈ wcel 2141  Vcvv 3453  ∅c0 4285   × cxp 5659  ‘cfv 6536  mTCcmtc 35922  mRExcmrex 35924  mExcmex 35925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-mex 35945

This theorem is referenced by:  mexval2  35961  msubff  35988  msubco  35989  msubff1  36014  mvhf  36016  msubvrs  36018